Question 131292


Looking at {{{35z^2-23z-72}}} we can see that the first term is {{{35z^2}}} and the last term is {{{-72}}} where the coefficients are 35 and -72 respectively.


Now multiply the first coefficient 35 and the last coefficient -72 to get -2520. Now what two numbers multiply to -2520 and add to the  middle coefficient -23? Let's list all of the factors of -2520:




Factors of -2520:

1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520


-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-12,-14,-15,-18,-20,-21,-24,-28,-30,-35,-36,-40,-42,-45,-56,-60,-63,-70,-72,-84,-90,-105,-120,-126,-140,-168,-180,-210,-252,-280,-315,-360,-420,-504,-630,-840,-1260,-2520 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -2520

(1)*(-2520)

(2)*(-1260)

(3)*(-840)

(4)*(-630)

(5)*(-504)

(6)*(-420)

(7)*(-360)

(8)*(-315)

(9)*(-280)

(10)*(-252)

(12)*(-210)

(14)*(-180)

(15)*(-168)

(18)*(-140)

(20)*(-126)

(21)*(-120)

(24)*(-105)

(28)*(-90)

(30)*(-84)

(35)*(-72)

(36)*(-70)

(40)*(-63)

(42)*(-60)

(45)*(-56)

(-1)*(2520)

(-2)*(1260)

(-3)*(840)

(-4)*(630)

(-5)*(504)

(-6)*(420)

(-7)*(360)

(-8)*(315)

(-9)*(280)

(-10)*(252)

(-12)*(210)

(-14)*(180)

(-15)*(168)

(-18)*(140)

(-20)*(126)

(-21)*(120)

(-24)*(105)

(-28)*(90)

(-30)*(84)

(-35)*(72)

(-36)*(70)

(-40)*(63)

(-42)*(60)

(-45)*(56)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -23? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -23


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-2520</td><td>1+(-2520)=-2519</td></tr><tr><td align="center">2</td><td align="center">-1260</td><td>2+(-1260)=-1258</td></tr><tr><td align="center">3</td><td align="center">-840</td><td>3+(-840)=-837</td></tr><tr><td align="center">4</td><td align="center">-630</td><td>4+(-630)=-626</td></tr><tr><td align="center">5</td><td align="center">-504</td><td>5+(-504)=-499</td></tr><tr><td align="center">6</td><td align="center">-420</td><td>6+(-420)=-414</td></tr><tr><td align="center">7</td><td align="center">-360</td><td>7+(-360)=-353</td></tr><tr><td align="center">8</td><td align="center">-315</td><td>8+(-315)=-307</td></tr><tr><td align="center">9</td><td align="center">-280</td><td>9+(-280)=-271</td></tr><tr><td align="center">10</td><td align="center">-252</td><td>10+(-252)=-242</td></tr><tr><td align="center">12</td><td align="center">-210</td><td>12+(-210)=-198</td></tr><tr><td align="center">14</td><td align="center">-180</td><td>14+(-180)=-166</td></tr><tr><td align="center">15</td><td align="center">-168</td><td>15+(-168)=-153</td></tr><tr><td align="center">18</td><td align="center">-140</td><td>18+(-140)=-122</td></tr><tr><td align="center">20</td><td align="center">-126</td><td>20+(-126)=-106</td></tr><tr><td align="center">21</td><td align="center">-120</td><td>21+(-120)=-99</td></tr><tr><td align="center">24</td><td align="center">-105</td><td>24+(-105)=-81</td></tr><tr><td align="center">28</td><td align="center">-90</td><td>28+(-90)=-62</td></tr><tr><td align="center">30</td><td align="center">-84</td><td>30+(-84)=-54</td></tr><tr><td align="center">35</td><td align="center">-72</td><td>35+(-72)=-37</td></tr><tr><td align="center">36</td><td align="center">-70</td><td>36+(-70)=-34</td></tr><tr><td align="center">40</td><td align="center">-63</td><td>40+(-63)=-23</td></tr><tr><td align="center">42</td><td align="center">-60</td><td>42+(-60)=-18</td></tr><tr><td align="center">45</td><td align="center">-56</td><td>45+(-56)=-11</td></tr><tr><td align="center">-1</td><td align="center">2520</td><td>-1+2520=2519</td></tr><tr><td align="center">-2</td><td align="center">1260</td><td>-2+1260=1258</td></tr><tr><td align="center">-3</td><td align="center">840</td><td>-3+840=837</td></tr><tr><td align="center">-4</td><td align="center">630</td><td>-4+630=626</td></tr><tr><td align="center">-5</td><td align="center">504</td><td>-5+504=499</td></tr><tr><td align="center">-6</td><td align="center">420</td><td>-6+420=414</td></tr><tr><td align="center">-7</td><td align="center">360</td><td>-7+360=353</td></tr><tr><td align="center">-8</td><td align="center">315</td><td>-8+315=307</td></tr><tr><td align="center">-9</td><td align="center">280</td><td>-9+280=271</td></tr><tr><td align="center">-10</td><td align="center">252</td><td>-10+252=242</td></tr><tr><td align="center">-12</td><td align="center">210</td><td>-12+210=198</td></tr><tr><td align="center">-14</td><td align="center">180</td><td>-14+180=166</td></tr><tr><td align="center">-15</td><td align="center">168</td><td>-15+168=153</td></tr><tr><td align="center">-18</td><td align="center">140</td><td>-18+140=122</td></tr><tr><td align="center">-20</td><td align="center">126</td><td>-20+126=106</td></tr><tr><td align="center">-21</td><td align="center">120</td><td>-21+120=99</td></tr><tr><td align="center">-24</td><td align="center">105</td><td>-24+105=81</td></tr><tr><td align="center">-28</td><td align="center">90</td><td>-28+90=62</td></tr><tr><td align="center">-30</td><td align="center">84</td><td>-30+84=54</td></tr><tr><td align="center">-35</td><td align="center">72</td><td>-35+72=37</td></tr><tr><td align="center">-36</td><td align="center">70</td><td>-36+70=34</td></tr><tr><td align="center">-40</td><td align="center">63</td><td>-40+63=23</td></tr><tr><td align="center">-42</td><td align="center">60</td><td>-42+60=18</td></tr><tr><td align="center">-45</td><td align="center">56</td><td>-45+56=11</td></tr></table>



From this list we can see that 40 and -63 add up to -23 and multiply to -2520



Now looking at the expression {{{35z^2-23z-72}}}, replace {{{-23z}}} with {{{40z+-63z}}} (notice {{{40z+-63z}}} adds up to {{{-23z}}}. So it is equivalent to {{{-23z}}})


{{{35z^2+highlight(40z+-63z)+-72}}}



Now let's factor {{{35z^2+40z-63z-72}}} by grouping:



{{{(35z^2+40z)+(-63z-72)}}} Group like terms



{{{5z(7z+8)-9(7z+8)}}} Factor out the GCF of {{{5z}}} out of the first group. Factor out the GCF of {{{-9}}} out of the second group



{{{(5z-9)(7z+8)}}} Since we have a common term of {{{7z+8}}}, we can combine like terms


So {{{35z^2+40z-63z-72}}} factors to {{{(5z-9)(7z+8)}}}



So this also means that {{{35z^2-23z-72}}} factors to {{{(5z-9)(7z+8)}}} (since {{{35z^2-23z-72}}} is equivalent to {{{35z^2+40z-63z-72}}})




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     Answer:

So {{{35z^2-23z-72}}} factors to {{{(5z-9)(7z+8)}}}



So one of the factors of {{{35z^2-23z-72}}} is D) {{{5z-9}}}